The concept of energy (and the conservation of energy) is one of the most important topics in physics. Work Dot produc

1. Chapter 7: Work and Kinetic Energy-1 Reading assignment: Chapter 7.6-7.9 Homework: due Monday, Sept 28, 2009 Chapter 7: 5AE's, 5AF's, Q7, 2, 9, 18, 22 • The concept of energy (and the conservation of energy) is one of the most important topics in physics. • Work • Dot products • Energy approach is simpler than Newton’s second law.

2. 1. A woman holds a bowling ball in a fixed position. The work she does on the ball ___ 1. depends on the weight of the ball. ___ 2. cannot be calculated without more information. ___ 3. is equal to zero. 2. A man pushes a very heavy load across a horizontal floor. The work done by gravity on the load ___ 1. depends on the weight of the load. ___ 2. cannot be calculated without more information. ___ 3. is equal to zero.

3. 3. When net positive work is done on a particle, its kinetic energy a. increases. b. decreases. c. remains the same. d. need more information about the way the work was done 4. In a collision between two billiard balls, a. energy is not conserved if the collision is perfectly elastic. b. momentum is not conserved if the collision is inelastic. c. not covered in the reading assignment

4. Work (as defined by a physicist) Definition: The work done on an object by an external force is - the product of the component of the force in the direction of the displacement and the magnitude of the displacement.

7. How much work is done when just holding up an object?

8. How much work is done when the displacement is perpendicular to the force?

9. F = 10 N • What is the work done by • the gravitational force • the normal force • the force F • when the block is displaced along the horizontal. q = 60° d = 10m The total work is: - the sum of the work done by all forces - or:

10. What is the work done when lifting? • By the gravitational force? • By the applied force? Strongest man lifting 140 kg boulder by 1 m. Sign convention: W is positive: If F and d are parallel If energy is transferred into the system W is positive: If F and d are antiparallel If energy is transferred out of the system

12. Work is a scalar quantitiy. (not a vector) Work has units of newton·meter (N·m) = the joule (J)

13. Black board example 7.5 A man loads a refrigerator onto a truck using a ramp. Ignore friction. He claims he would be doing less work if the length of the ramp would be longer. Is this true?

15. 500 N Black board example 7.1 A donkey is pulling a cart with a force of magnitude F = 500 N at an angle of 30º with the horizontal. Calculate the work done by the donkey as the cart is pulled for one mile (1648 m).

16. Definition of dot product and work Work is the scalar product (or dot product) of the force F and the displacement d. F and d are vectors W is a scalar quantity

17. Scalar product between vector A and B Definition: Scalar product is commutative: Distributive law of multiplication:

18. Scalar Product using unit vectors: We have the vectors A and B: Then:

19. Black board example 7.2 • A particle moving in the x-y plane undergoes a displacement d = (2.0i + 3.0j) m at a constant force F = (5.0i + 2.0j) N acts on the particle. Calculate • The magnitude of the displacement and the force. • The work done by F. • The angle between F and d.

23. 1. The gravitational potential energy of a particle at a height z above Earth’s surface ___ 1. depends on the height z. ___ 2. depends on the path taken to bring the particle to z. ___ 3. both 1 and 2. ___ 4. is not covered in the reading assignment. 2. Which of the following is not a conservative force? ___ 1. the force exerted by a spring on a particle in one dimension ___ 2. the force of friction ___ 3. the force of gravity ___ 4. not covered in the reading assignment 3. Which of the following was not discussed in the reading assignment? ___ 1. conservation of non-conservative forces ___ 2. block and tackle ___ 3. work ___ 4. all of the above were discussed

24. Black board example 7.2 • A particle moving in the x-y plane undergoes a displacement d = (2.0i + 3.0j) m at a constant force F = (5.0i + 2.0j) N acts on the particle. Calculate • The magnitude of the displacement and the force. • The work done by F. • The angle between F and d.

25. What if the force varies? We have to integrate the force along x Work done by a varying force: Thus, the work is equal to the area under the F(x) vs. x curve.

26. Black board example 7.3 A force acting on a particle varies as shown in the Figure. What is the total work done on the particle as it is moved from x = 0 to x = 8 m? Hint: It is the area under the curve.

27. Consider a spring Hooke’s law: (Force required to stretch or compress a spring by x): k is the spring constant of a spring. Stiff springs have a large k value.

28. Work done by a spring xi xf

29. A spring-loaded toy dart gun is used to shoot a dart straight up in the air, and the dart reaches a maximum height of 24 m.The same dart is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the dart go this time, neglecting friction and assuming an ideal spring? 1. 96 m 2. 48 m 3. 24 m 4. 12 m 5. 6 m 6. 3 m 7. impossible to determine

30. Black board example 7.3 A 0.500 kg mass is hung from a spring extending the spring by a distance x = 0.2 m • What is the spring constant of the spring? • How much work was done on the mass by the gravitational force • How much work was done on the mass by the spring force?

31. The kinetic energy of a particle is:

32. A bullet of mass m = 0.020 kg moves at 500 m/s. A truck of mass m = 1000 kg moves at 5 m/s Which has more kinetic energy?

33. Work due to friction If friction is involved in moving objects, work has to be done against the kinetic frictional force. This work is:

34. A cart on an air track is moving at 0.5 m/s when the air is suddenly turned off. The cart comes to rest after traveling 1m. The experiment is repeated, but now the cart is moving at 1 m/s when the air is turned off. How far does the cart travel before coming to rest? 1. 1 m 2. 2 m 3. 3 m 4. 4 m 5. 5 m 6. impossible to determine

35. Black board example 7.4 • Angus is pulling a 10,000 kg truck with all his might (2000N) on a frictionless surface for 10.0 m. • How much work is the man doing? • What is the speed of the truck after 10 m. • What is the speed of the truck after 10 m if there is friction? • (friction coefficient: 0.0153)

36. Power Power is the rate at which work is done: Average power: (work done per time interval Dt)

37. The power can also be expressed as: (Dot product) The units of power are joule/sec (J/s) = watt (W)

38. Black board example 7.7 • An elevator having a total mass of 3000 kg moves upward against the gravitational force at a constant speed of 9.13 m/s. • What is the power delivered by the motor?

40. Chapter 8: Potential Energy and Conservation of Energy part 1 (finish Chp7) Reading assignment: Chapter 8.5-8-7 Homework : due Monday, October 5, 2009 Problems: Chapter 8 33, 36, 37, 58, 65 Bonus: 48, 64, 37 • One form of energy can be converted into another form of energy. • Conservative and non-conservative forces • CONSERVATION OF ENERGY

41. 1. Suppose you know the potential energy function corresponding to a force. Is it always possible to calculate the force? ___ 1. yes ___ 2. only if the force is nonconservative ___ 3. not covered in the reading assignment 2. The potential energy of a spring is ___ 1. proportional to the amount the spring is stretched. ___ 2. proportional to the square of the amount the spring is stretched. ___ 3. not yet covered in any reading assignment. 3. A car slows down as a result of air friction. Which is true? ___ 1. The car’s kinetic energy decreases. ___ 2. Heat is generated. ___ 3. The energy of the car/road/air system is constant. ___ 4. all of the above ___ 5. none of the above

42. Potential energy U: • Can be thought of as stored energy that can either do work or be converted to kinetic energy. • When work gets done on an object, its potential and/or kinetic energy increases. • There are different types of potential energy: • Gravitational energy • Elastic potential energy (energy in an stretched spring) • Others (magnetic, electric, chemical, …)

43. Conservative and non-conservative forces Conservative forces: Work is independent of the path taken. Work depends only on the final and initial point. Work done is zero if the path is a closed loop (same beginning and ending points.) We can always associate a potential energy with conservative forces. We can only associate a potential energy with conservative forces. Work done by a conservative force: Wc = Ui – Uf = - DU Examples of conservative forces: _____________________________________________

44. Conservative forces and potential energy The work done by a conservative force equals the negative of the change in potential energy associated with that force. Any conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x.

45. Conservative and non-conservative forces Non-conservative forces: • A force is non-conservative if it causes a change in mechanical energy; mechanical energy is the sum of kinetic and potential energy. • Example: Frictional force. • This energy cannot be converted back into other forms of energy (irreversible). • Work does depend on path. Sliding a book on a table

46. Gravitational potential energy: - Potential energy only depends on y (height) and not on x (lateral distance)

47. Black board example 8.1 • You are 1.80 m tall. • A 0.1 kg apple, which is hanging 1 m above your head, drops on you. • What is the difference in gravitational potential energy when it hangs and when it hits you? • How much gravitational potential energy does it loose? 1 m

48. Work done by/on a spring: xi xf Elastic potential energy stored in a spring: The spring is stretched or compresses from its equilibrium position by x

49. Review Important energy formulas: Work: Forms of energy:

50. Demo example (conversion of energy): (ballistic pendulum) Conversion of: Elastic potential energy into kinetic energy into gravitational potential energy